Foundations and Applications

Atomic Nucleus

Photo by: Petr Vaclavek

The atomic nucleus is a tiny massive entity at the center of an atom.
Occupying a volume whose radius is 1/100,000 the size of the atom, the
nucleus contains most (99.9%) of the mass of the atom. In describing the
nucleus, we shall describe its composition, size, density, and the forces
that hold it together. After describing the structure of the nucleus, we
shall go on to describe some of the limits of
nuclear
stability.

The nucleus is composed of protons (charge = +1; mass = 1.007
atomic mass units
([μ]) and neutrons. The number of protons in the nucleus is called
the
atomic number
Z and defines which chemical element the nucleus represents. The number
of neutrons in the nucleus is called the neutron number N, whereas the
total number of neutrons and protons in the nucleus is referred to as the
mass number A, where A = N + Z. The neutrons and protons are referred to
collectively as nucleons. A nucleus with a given N and Z is referred to as
a nuclide. Nuclides with the same atomic number are
isotopes
, such as
12
C and
14
C, whereas nuclides with the same N, such as
14
C and
16
O, are called isotones. Nuclei such as
14
N and
14
C, which have the same mass number, are isobars. Nuclides are designated
by a shorthand notation in which one writes
, that is, for a nucleus with 6 protons and 8 neutrons, one writes
, or,
, or just
14
C. The size of a nucleus is approximately 1 to 10 × 10
−15
m, with the nuclear radius being represented more precisely as 1.2
× A
1/3
× 10
−15
m. We can roughly approximate the nucleus as a sphere and thus we can
calculate its density

where 1.66 × 10
−27
kg is the mass of the nucleon. Thus the nuclear density is about 200,000
tonnes/mm
3
and is independent of A. Imagine a cube that is 1 mm on a side. If filled
with nuclear matter, it would have a mass of about 200,000 tonnes. This
calculation demonstrates the enormous matter/energy density of nuclei and
gives some idea as to why nuclear phenomena lead to large energy releases.

Of the 6,000 species of nuclei that can exist in the universe, about 2,700
are known, but only 270 of these are stable. The rest are radioactive,
that is, they spontaneously decay. The driving force behind all
radioactive decay
is the ability to produce products of greater stability than one had
initially. In other words, radioactive decay releases energy and because
of the high energy density of nuclei, that energy release is substantial.
Qualitatively we describe radioactive decay as occurring in three general
ways:
α
-,
β
-, and
γ
-decay. Alpha-decay occurs in the heavy elements, and consists of the
emission of a
4
He nucleus. Beta-decay occurs in nuclei whose N/Z ratio is different from
that of a stable nucleus and consists of a transformation of neutrons into
protons or vice versa to make the nucleus more stable. Gamma-decay occurs
when excited nuclei get rid of some or all of their excitation energy via
the emission of electromagnetic radiation, or via the radiationless
transfer of energy to orbital electrons.

The force responsible for holding the neutrons and protons together within
the very small nuclear volume must be unusually strong. The nuclear force,
or strong interaction, is one of the four fundamental forces of nature
(namely, the gravitational, electromagnetic, strong, and weak forces). The
nuclear force is charge-independent, meaning that the nuclear force
between two protons, or two neutrons, or a neutron and a proton, is the
same. The nuclear force is short-ranged, meaning it acts over a distance
of 10
−15
to 10
−14
m, that is, the size of nuclei. Of course the nuclear force is
attractive, as it binds the nucleons in a nucleus. But some experiments
have shown the nuclear force has a "repulsive core," meaning
that at very short distances,
the force switches from attractive to repulsive, preventing the nucleus
from collapsing in on itself. The nuclear force is an
"exchange" force, resulting from the virtual exchange of
pions (short-lived particles with integral spin, produced normally in
nuclear reactions) between interacting nucleons. More recently we have
come to understand the nuclear force using a theory called quantum
chromodynamics (QCD), which describes protons and neutrons as being made
up of quarks. In particular, the proton is thought of as a combination of
two up quarks (uu) and a down quark (d), whereas the neutron is thought to
consist of one up quark (u) and two down quarks (dd). The up and down
quarks are small particles with charges of +2/3 and −1/3,
respectively. The quarks account for approximately 2 percent of the mass
of the proton. The rest of the mass consists of gluons, which are the
particles exchanged between the quarks to bind them together. The force
acting between the quarks has the unusual property of being small when
they are close together, and increasing as the distance between them
grows. Because of this, no isolated quarks have been observed in nature.

In close analogy to atomic structure, we speak of the structure of various
nuclei. Many nuclear properties can be described using a nuclear shell
model in which the nucleons are placed in orbitals like electrons in
atoms. These orbitals and their properties are predicted by applying
quantum mechanics to the problem of defining the states of the nucleons,
which move under the influence of the average force in the nucleus. Like
atoms, there are certain configurations of nucleons that have special
stability, for example, the so-called magic numbers akin to the
inert
gas structures in chemistry. In addition to those nuclear properties that
are best described in terms of a shell model, there are other properties
that seem to be best explained by the large-scale collective motion of a
number of nucleons. These motions lead to nuclear rotations and
vibrations, which are described by a nuclear collective model.

Current research on nuclei, their properties, and the forces that hold
them together focuses on studying nuclei at the limits of stability. The
basic idea is that when one studies nuclei under extreme conditions, one
then has a unique ability to test theories and models that were designed
to describe the "normal" properties of nuclei. One limit of
nuclear stability is that of high Z, that is, as the atomic number of the
nucleus increases, the repulsion between the nuclear protons becomes so
large as to cause the nucleus to spontaneously
fission
. The competition between this repulsive Coulomb force and the cohesive
nuclear force is what defines the size of the Periodic Table and the
number of chemical elements. At present there are 112 known chemical
elements, and evidence for the successful
synthesis
of elements having the atomic numbers 114 and 116 has been presented.

Another limit of nuclear stability is the extreme of the neutron to proton
ratio, N/Z. For certain very neutron-rich nuclei, such as
11
Li, an unusual halo structure has been observed. In halo nuclei, a
"core" of nucleons is surrounded by a "misty cloud, a
halo" of
valence
nucleons that are weakly bound and extend out to great distances,
analogous to electrons surrounding the nucleus in an atom. Halo nuclei are
fragile objects, are relatively large, and interact easily with other
nuclei (have enhanced reaction cross sections). The halo nucleus
11
Li, which has a
9
Li core surrounded by a two-neutron halo is shown in Figure 1.
11
Li is as large as
208
Pb.
11
Li and other

Figure 1. Schematic views of the nuclear halo nuclei
11
Li and
19
C that compares them to
208
Pb and the Borromean rings of medieval times. (Source:
http://www.phy.anl.gov/ria/index)

two-neutron halo nuclei are three-body systems (2 neutrons and a
9
Li core), which pose a special challenge to nuclear theorists. They are
also examples of Borromean systems, in which the nucleus is no longer
bound if any one of the three components is removed. (The name derives
from the heraldic emblem of medieval princes of Borromeo, which has three
rings interlocked in such a way that removal of any one ring will make the
others fall apart.)

There appear to be missing links in the explanation:
1. Please define the size of the Nuleus. You have given it unit of length. In other words, what exactly does it represent
2. you have given the formula of radius, too. Please explain how did you arrive at it.
3. if Mass number, A, is part of radius formula, why it does not play any role in calculation of volume. Please explain.